The autofocus problem in synthetic aperture radar imaging amounts to estimating unknown phase errors caused by unknown platform or target motion. At the heart of three state-of-the-art autofocus algorithms, namely, phase gradient autofocus, multichannel autofocus (MCA), and Fourier-domain multichannel autofocus (FMCA), is the solution of a constant modulus quadratic program (CMQP). Currently, these algorithms solve a CMQP by using an eigenvalue relaxation approach. We propose an alternative relaxation approach based on semidefinite programming, which has recently attracted considerable attention in other signal processing problems. Experimental results show that our proposed methods provide promising performance improvements for MCA and FMCA through an increase in computational complexity.

Autofocus algorithms are used to restore images in nonideal synthetic aperture radar imaging systems. In this paper, we propose a bilinear parametric model for the unknown image and the nuisance phase parameters and derive an efficient maximum-likelihood autofocus (MLA) algorithm. In the special case of a simple image model and a narrow range of look angles, MLA coincides with the successful multichannel autofocus (MCA). MLA can be interpreted as a generalization of MCA to a larger class of models with a larger range of look angles. We analyze its advantages over previous extensions of MCA in terms of identifiability conditions and noise sensitivity. As a byproduct, we also propose numerical approximations to the difficult constant modulus quadratic program that lies at the core of these algorithms. We demonstrate the superior performance of our proposed methods using computer simulations in both the correct and mismatched system models. MLA performs better than other methods, both in terms of the mean squared error and visual quality of the restored image.

Autofocus algorithms deal with image restoration in a nonideal synthetic aperture radar (SAR) imaging system. We propose a novel autofocus algorithm, denoted as MLA, that is based on maximum likelihood estimation. MLA belongs to a class of autofocus algorithms that rely on a known low-return region in the underlying image. We find conditions under which MLA is equivalent to previous methods belonging to the same class. Simulation results show that when compared to previous methods, MLA performs better both in terms of visual quality of the restored image and mean square error (MSE) of the estimated unknown parameters.

Synthetic aperture radar (SAR) imaging suffers from image focus degradation in the presence of phase errors in the received signal due to unknown platform motion or signal propagation delays. We present a new autofocus algorithm, termed Fourier-domain multichannel autofocus (FMCA), that is derived under a linear algebraic framework, allowing the SAR image to be focused in a noniterative fashion. Motivated by the mutichannel autofocus (MCA) approach, the proposed autofocus algorithm invokes the assumption of a low-return region, which generally is provided within the antenna sidelobes. Unlike MCA, FMCA works with the collected polar Fourier data directly and is capable of accommodating wide-angle monostatic SAR and bistatic SAR scenarios. Most previous SAR autofocus algorithms rely on the prior assumption that radar's range of look angles is small so that the phase errors can be modeled as varying along only one dimension in the collected Fourier data. And, in some cases, implicit assumptions are made regarding the SAR scene. Performance of such autofocus algorithms degrades if the assumptions are not satisfied. The proposed algorithm has the advantage that it does not require prior assumptions about the range of look angles, nor characteristics of the scene.

MultiChannel Autofocus (MCA) and Reversed-step MCA (RMCA) for Synthetic Aperture Radar (SAR) assume a region of low return in the focused data, and work best when the region is known accurately. In practice, the returns from within the side-lobes of the antenna footprint are highly attenuated; thus a region of highly-probable low returns is provided. However, scene-dependent features like strong reflectors in the presumed low-return region may violate the assumptions underlying MCA and RMCA, with resulting performance degradation. We consider such a scenario and propose scene-independent, random selection methods of the low-return constraints. Our simulation results show that a well-focused image can be obtained within a few trials, even when the number of anomalies is moderately large.

Synthetic Aperture Radar (SAR) imaging can suffer from image focus degradation due to unknown platform or target motion. Autofocus algorithms use signal processing techniques to remove the undesired phase errors. The recently proposed multichannel autofocus models formulate the problem as the solution to Aejφ≈ 0, where A is a given matrix and φ are the unknown phases. Previous methods approximated ejf using the null vector of A. We propose to approximate ejφusing conic optimization and call this new autofocus algorithm Semidefinite Relaxation Autofocus (SDRA). Experimental results using a simulated SAR image shows that SDRA has promising performance advantages over existing autofocus methods.

We present a new noniterative approach to synthetic aperture radar (SAR) autofocus, termed the multichannel autofocus (MCA) algorithm. The key in the approach is to exploit the multichannel redundancy of the defocusing operation to create a linear subspace, where the unknown perfectly focused image resides, expressed in terms of a known basis formed from the given defocused image. A unique solution for the perfectly focused image is then directly determined through a linear algebraic formulation by invoking an additional image support condition. The MCA approach is found to be computationally efficient and robust and does not require prior assumptions about the SAR scene used in existing methods. In addition, the vector-space formulation of MCA allows sharpness metric optimization to be easily incorporated within the restoration framework as a regularization term. We present experimental results characterizing the performance of MCA in comparison with conventional autofocus methods and discuss the practical implementation of the technique.

Synthetic Aperture Radar (SAR) imaging suffers from image focus degradation in the presence of phase errors in the received signals due to unknown platform or target motion. We study automatic focusing (autofocus) under a multistatic passive scenario, where the illumination is provided by a set of stationary UHF transmitters and the goal is to image a moving aircraft. We employ heuristic iterative estimation algorithms that maximize a sharpness metric of the image. A similar method has been studied for the case of mono-static radar, where only one antenna is used for both transmitting and receiving. We present simulation results to help assess the effectiveness of the proposed autofocus approach.

Synthetic aperture radar (SAR) autofocus techniques that optimize sharpness metrics can produce excellent restorations in comparison with conventional autofocus approaches. To help formalize the understanding of metric-based SAR autofocus methods, and to gain more insight into their performance, we present a theoretical analysis of these techniques using simple image models. Specifically, we consider the intensity-squared metric, and a dominant point-targets image model, and derive expressions for the resulting objective function. We examine the conditions under which the perfectly focused image models correspond to stationary points of the objective function. A key contribution is that we demonstrate formally, for the specific case of intensity-squared minimization autofocus, the mechanism by which metric-based methods utilize the multichannel defocusing model of SAR autofocus to enforce the stationary point property for multiple image columns. Furthermore, our analysis shows that the objective function has a special separble property through which it can be well approximated locally by a sum of 1-D functions of each phase error component. This allows fast performance through solving a sequence of 1-D optimization problems for each phase component simultaneously. Simulation results using the proposed models and actual SAR imagery confirm that the analysis extends well to realistic situations.

Phase unwrapping is an important problem in many magnetic resonance imaging applications, such as field mapping and flow imaging. The challenge in two-dimensional phase unwrapping lies in distinguishing jumps due to phase wrapping from those due to noise and/or abrupt variations in the actual function. This paper addresses this problem using a Markov random field to model the true phase function, whose parameters are determined by maximizing the a posteriori probability. To reduce the computational complexity of the optimization procedure, an efficient algorithm is also proposed for parameter estimation using a series of dynamic programming connected by the iterated conditional modes. The proposed method has been tested with both simulated and experimental data, yielding better results than some of the state-of-the-art method (e.g., the popular least-squares method) in handling noisy phase images with rapid phase variations.

We propose a new fast Feldkamp algorithm for 3-D cone beam tomography with a circular source trajectory. The algorithm is an extension of our recent fast native 2-D fan-beam reconstruction algorithm. It is based on a recursive hierarchical decomposition of the cone-beam backprojection operation into successively smaller sub-volumes. The algorithm reduces the computational complexity of the reconstruction from O(N/sup 4/) to O(N/sup 3/ log N). Simulations demonstrate the efficiency of our algorithm, with 7-fold speedup for a 128 /spl times/ 128 /spl times/ 128 image. Speedups will be much greater for images of more typical size encountered in medicine.

When estimating terrain elevation via interferometric synthetic aperture radar (InSAR), phase unwrapping procedures have difficulty in dealing with rough regions or large noise. Multiple baseline is used to reduce or avoid this problem. Conventional maximum likelihood (ML) methods reconstruct terrain heights in a pointwise fashion, which does not utilize the smooth characteristics of natural terrain. We propose a new algorithm taking smoothness into account. The new approach tackles the problem in a Bayesian framework. Instead of using ML estimation, we use maximum a posteriori (MAP) estimation, where the likelihood function is defined as in the ML method and the prior is defined as a first-order Gaussian Markov random field. This MAP estimation makes the algorithm more robust to noise, and at the same time, more accurate in reconstructing rough regions. A form of 2-D dynamic programming is used to implement the MAP estimation efficiently. The new algorithm has the advantage over the ML methods in that none of the baselines must be chosen so small as to avoid phase wrapping. Specifically, both baselines can be large so that the noise in the reconstructed height can be low. The new algorithm is shown to be able to achieve lower noise than the conventional ML and least-squares methods.

Imaging of moving targets using synthetic aperture radar (SAR) is a challenging task, due to unknown motion of the targets. In recent years, time-frequency (T-F) methods, based on the Wigner-Ville distribution (WVD), have been proposed to form high resolution image sequences of moving targets. Previously, we analyzed this approach, assuming targets with constant velocity, by deriving an approximate closed-form expression for the point-target response of the WVD-based SAR imaging system. In this paper, we extend our analyses to targets having constant acceleration. Then we address several other issues associated with this WVD-based SAR imaging technique. Simulation results are presented to validate our analysis.

Using a far-field model, bistatic synthetic aperture radar (SAR) acquires Fourier data on a rather unusual, non-Cartesian grid in the Fourier domain. Previous image formation algorithms were mainly based on direct Fourier reconstruction to take advantage of the FFT, but the irregular coverage of the available Fourier domain data and the 2-D interpolation in the Fourier domain may adversely affect the accuracy of image reconstruction. Back-projection techniques avoid Fourier-domain interpolation, but ordinarily have huge computational cost. We present a fast back-projection algorithm for bistatic SAR imaging, motivated by a fast back-projection algorithm previously proposed for tomography. It has a reduced computational cost, on the same order as that of direct Fourier reconstruction. Furthermore, this approach can be used for near-field imaging. Simulation results verify the performance of this new algorithm.

The stage by stage approaching (SSA) entropy minimization approach to synthetic aperture radar (SAR) autofocus is a modern and innovative technique, which, to date, has not been compared with other approaches. We provide such a comparison and show that SSA is a most promising autofocus method. We evaluate the merit of this algorithm through comparison with the established phase gradient autofocus (PGA) algorithm on both simulated and real SAR data. Our results show that SSA works well in comparison to the state-of-the-art PGA. However, SSA is more computationally demanding.

We consider a novel Bayesian approach to 2-D phase unwrapping. The phase is unwrapped according to a maximum a posteriori (MAP) rule, where the estimate is made through a form of 2-D dynamic programming. The approach uses structured iterated conditional modes to achieve good performance without examining a large number of states in the dynamic system. We analyze the performance of the approach by transforming the problem to one of decoding a convolutional code. An example with seven states in the dynamic program is given. We derive an approximate upper bound for the probability of pixel error based on a Gaussian Markov random field model. Monte Carlo simulation results show that the bound offers a good approximation to the probability of error. A comparison with other phase unwrapping techniques on a real data set suggests that the new approach is superior.

Phase unwrapping in 2-dimensional topologies is an important problem that has several applications in radar and satellite imaging. The s um product algorithm (belief propagation) gives excellent results for the phase unwrapping problem. In this work, we present a gradient smoothing technique that uses higher order surface models to produce very smooth surfaces and report an improvement in the solution obtained. In a recent important work, Yedidia et. al have showed the theoretical connections between belief propagation algorithms and free energy in statistical physics. Based on this, we present a model that uses the Kikuchi technique to compute better posterior marginals than those produced by sum product algorithm.

Describes two classes of two-dimensional phase unwrapping algorithms. It is shown that combining advantages from each of these classes is useful in improving phase-unwrapping solutions.

Abstract — We propose a novel Bayesian approach to 2-D phase unwrapping. Modeled as a first-order Gaussian Markov random field, the unwrapped phase is estimated according to a maximum a posteriori (MAP) rule. The estimate is made through a form of 2-D dynamic programming, using a series of row-by-row or column-by-column 1-D dynamic programming optimiza-tions. Increasing the number of states in the dynamic system can improve the unwrapping performance, but also increases the com-putational complexity. Due to this trade-off, a structured iterated conditional mode (SICM) is used to achieve good performance without examining a large number of states in each iteration. A row-by-row followed by column-by-column raster scan takes pre-vious estimates into account through a weighting. Other raster scans are also possible. The approach can be implemented effi-ciently in terms of memory usage due to the recyclable memory of dynamic programming. An example of the approach with seven states is given. Experimental results are compared to other algo-rithms including the least-squares method, the branch-cut method and Flynn’s method, using interferometric SAR data. The new SICM algorithm is seen to be superior. I.

Some types of medical and topographic imaging devices produce images in which the pixel values are "phase-wrapped", i.e., the measured modulus is a known scalar. Phase unwrapping can be viewed as the problem of inferring the number of shifts between each and every pair of neighboring pixels, subject to an a priori preference for smooth surfaces, and subject to a zero curl constraint, which requires that the shifts must sum to 0 around every loop. We formulate phase unwrapping as a mean field inference problem in a probability model, where the prior favors the zero curl constraint. We compare our mean field technique with the least squares method on a synthetic 100/spl times/100 image, and give results on a larger 512/spl times/512 image.

Phase images are derived from source images by applying a modulus operation to each pixel value. Phase unwrapping is the problem of inferring the original, unwrapped values from the wrapped values, using prior knowledge about the smoothness of the image. One approach to solving this problem is to infer the gradient vector field of the unwrapped image and then integrate the gradient field. The gradient in a particular direction at a pixel is equal to the observed pixel difference plus an unknown integer number of shifts. We introduce a technique for inferring these shifts using the low-complexity probability propagation algorithm, applied in a graphical model that prefers shifts that match the phase image and that constrains the shifts to satisfy the properties of a gradient field. We present results for a phase image from the region of the Sandia National Laboratories.

This paper outlines the goals, structure, and technology elements of the INFINITY Project, a joint effort between university educators, industrial partners, and civic leaders to introduce a signal-processing-based engineering curriculum at the high school educational level. Implementation issues of the program are addressed, including class textbook and laboratory creation, teacher training, and online classroom support. Initial responses to the effort from teachers and students at fourteen different high schools are highly positive, and plans for further expansion of the program are given.

Detection and localization of underwater mines remains a challenging and important problem for safe operation of naval platforms. A number of new technologies exploit airborne LIDARs, which can penetrate the air-water interface and optically detect and localize underwater mines. Such systems process the received optical field generated by scattering within the water column, and have proven to be an effective technology for mine detection and localization. We consider the use of multiple looks at a single target to form a three-dimensional representation of the scatterers within the water column. To form such images, we account for the integration within the receive sensors, and formulate the problem in a tomographic framework. We present preliminary image formation results generated from data collected at sea with a state-of-the-art navy mine imaging system.

We investigate passive radar imaging of aircraft using reflected TV signals. We apply a smoothed pseudo Wigner-Ville distribution (SPWVD) based SAR imaging algorithm to two different scenarios. In the first simulation, a multistatic VHF-band dataset generated by fast Illinois solver code (FISC) is used. In the second simulation, a more realistic simulated passive radar dataset is used. A set of instantaneous images is produced by our algorithm. They have higher resolution and show more detail and features of the aircraft than can be obtained by direct Fourier reconstruction (DFR). The set of images provides more visual information about the target and helps to estimate its shape and features. This study suggests that the SPWVD-based imaging might be useful in passive radar imaging and target classification.

The wavenumber-domain SAR algorithm is studied for the case of a high squint angle with particular application to runway imaging from an approaching aircraft. The feasibility of using the algorithm for this application is shown by point target simulations which indicate that aberrations due to the high squint angle can be eliminated with finer interpolation in the processing.

We consider approximation of the optimal Yen algorithm (1956) for interpolation from a nonuniformly-spaced grid. Although the Yen interpolator is optimal in many senses, it suffers from severe numerical ill conditioning. We suggest a tradeoff between accuracy in computing the interpolator and accuracy in performing the interpolation. A new interpolator is proposed using Choi's expression (1998) for interpolation error. A strategy is suggested to control the error tradeoff. We also generalize the new interpolator to multiple dimensions. The newly designed sinc-kernel interpolator is compared with the Yen, Choi, and usual sinc interpolator with Jacobian weighting using simulations in both one and two dimensions. We show that the new interpolator is robust. It performs similarly to the Yen algorithm when noise is small and similarly to the Choi algorithm when noise is large.

Wavefront curvature must be taken into account in highly squinted SAR applications, such as runway imaging from an aircraft approaching for landing. The /spl omega/-k algorithm, which successfully incorporates wavefront curvature into the processing, shows certain aberrations in the highly squinted case, depending on the accuracy of the interpolation and the location of the targets in the scene. An alternative SAR inversion method performs a more general basis decomposition of the imaging kernel in order to accurately model the wavefront curvature. This general inversion method is inherently robust to noise, however, it is computationally much more expensive. The performance of the /spl omega/-k algorithm can be improved by increasing the computational load only linearly. Here we make a fair comparison of the two methods, where they are constrained to use the same amount of computation.

Sidelobe artifacts are a common problem in image reconstruction from finite-extent Fourier data. Conventional shift-invariant windows reduce sidelobe artifacts only at the expense of worsened mainlobe resolution. Spatially variant apodization (SVA) was previously introduced as a means of reducing sidelobe artifacts, while preserving mainlobe resolution. Although the algorithm has been shown to be effective in synthetic aperture radar (SAR), it is heuristically motivated and it has received somewhat limited analysis. In this paper, we show that SVA is a version of minimum-variance spectral estimation (MVSE). We then present a complete development of the four types of two-dimensional SVA for image reconstruction from partial Fourier data. We provide simulation results for various real-valued and complex-valued targets and point out some of the limitations of SVA. Performance measures are presented to help further evaluate the effectiveness of SVA.

This paper considers the problem of interpolating a signal from one uniformly-spaced grid to another, where the grid spacings may be related by an arbitrary, irrational factor. Noting that interpolation is the digital equivalent of magnification, we begin by reviewing optical systems for magnification and "computation" of the chirp Fourier transform. This route suggests several analog schemes for magnification, which can be discretized to produce algorithms for interpolation. We then derive one of these algorithms from first principles, using a digital signal processing perspective. The result is an important, but forgotten, algorithm for interpolation first suggested as an application of the chirp-z transform by Rabiner, Schafer, and Rader (1969). Unlike the earlier derivation, our approach is direct-we do not make use of Bluestein's trick of completing the square. In addition, our approach identifies parameters under user control that can be optimized for best performance.

The Department of Electrical and Computer Engineering at the University of Illinois recently adopted new undergraduate curricula. The most radical change was the introduction of ECE 210, Analog Signal Processing, in place of both the sophomore-level circuit analysis course and the junior-level signals and systems course. The new course combines core material from these traditional courses, along with applications such as AM radio and a modest laboratory component, in a way that improves both the students' understanding and their motivation. The new course still serves well as the base of the required curriculum and as a prerequisite for subsequent courses, while realizing savings in the early curriculum and allowing more time for advanced signal processing and systems courses in future semesters.

We develop an approach to the determination of 3-D structure and reflectivity of Lambertian objects using a modification of cone-beam tomography. Like other tomographic imaging methods, our approach is entirely numerical and does not rely on heuristics. Furthermore, it requires only ambient illumination.

A high-resolution acoustic imaging system is proposed for imaging objects buried in soil. The system incorporates a transducer array that collects acoustic data as it is moved across the soil surface. A mathematical model and associated computer software are developed in order to simulate the signals acquired by the actual system. Subsurface images are reconstructed from simulated and real data by using delay-and-sum beam-forming and by the application of synthetic aperture radar (SAR) theory to this particular problem.

We consider the application of a spotlight-mode synthetic aperture radar (SAR) imaging technique to the problem of high-resolution lunar imaging and other related radar astronomy problems. This approach offers improved image quality, compared with conventional processing, at the expense of slightly increased computational effort. Results of the processing of lunar data acquired with the 12.6 cm wavelength radar system at Arecibo Observatory are presented, and compared with the best available published result, by Stacy (1993), which uses focusing techniques from stripmap SAR.

A new undergraduate curriculum in electrical engineering has been adopted by the Department of Electrical and Computer Engineering at the University of Illinois. Major changes have been incorporated, including a redistribution of the circuits and signal processing topics within the curriculum. After giving an overview of the new curriculum, this paper focuses on a new, required sophomore-level course on analog signal processing. This course combines material from the traditional course on circuit analysis with material on continuous-time signals and systems. Students completing this course can study digital signal processing as first-semester juniors, which leaves ample time for more advanced signal and image processing courses in future semesters.

We explore 3-D image reconstruction from limited synthetic aperture radar (SAR) data, using a previously developed approach, based on high-resolution spectral estimation. Existing 3-D SAR imaging techniques, such as stereo and interferometry, present difficulties that may be overcome by this method. We use XPATCH to generate simulated SAR data of a military tank and an aircraft, and we then show image reconstructions using both conventional Fourier inversion and high-resolution spectral estimation. The latter approach is seen to provide superior imagery.

This article formally defines partial Radon transforms for functions of more than two dimensions. It shows that a generalized projection-slice theorem exists which connects planar and hyperplanar projections of a function to its Fourier transform. In addition, a general theoretical framework is provided for carrying out n-dimensional backprojection reconstruction in a multistage fashion through the use of the partial Radon transform.

We consider the application of an alternative imaging algorithm to the inversion of strip-mapping synthetic aperture radar (SAR) data. The algorithm was originally developed and successfully applied in the area of geophysics to estimate the conductivity distribution from wellbore induction measurements. The SAR measurement relation satisfies the same form of integral equation describing the wellbore problem. By exploiting the form of the measurement kernel, we derive a SAR image formation algorithm involving deconvolution-backprojection. Unlike correlation-based SAR image formation, our approach is more general, without simplifying assumptions on the range function, and is robust to measurement noise, at the expense of increased computational complexity. Simulation results are presented that demonstrate the effectiveness of the proposed algorithm.

We consider high-resolution radar imaging of an airport runway and environs from an approaching aircraft. Such a radar could provide successful imaging in all-weather conditions. We show the feasibility of using synthetic aperture radar (SAR) to achieve high-resolution reflectivity imagery of a runway region. A formula is given specifying the data collection time required to achieve a desired cross-range resolution. Traditional polar-format processing is inaccurate for this scenario because the required plane-wave assumption is violated. We explore two alternative approaches to SAR image formation based on the exact form of the range function: a general SAR inversion algorithm and the w-k algorithm. We present simulation results for these algorithms, based on realistic SAR parameters.

Studies the problem of interpolating a bandlimited signal from its nonuniform samples. The authors consider a class of interpolation algorithms that includes the least-squares optimal interpolator proposed by J.L. Yen, and derive a closed-form expression of the interpolation error for interpolators of this type. The expression for the interpolation error shows that the error depends on the eigenvalue distribution of a matrix, which is specified by the set of sampling points. The authors notice that the usual sinc-kernel interpolator is an approximation to the Yen interpolator, and suggest a method of choosing the weighting coefficients in the sinc-kernel interpolator. The new sinc-kernel interpolator is superior to the sinc interpolator with the usual Jacobian (area) weighting and is far easier to implement than the Yen interpolator.

Considers the problem of spotlight-mode synthetic aperture radar (SAR) imaging for an arbitrary radar path. A general imaging scenario involves a 3-D scattering surface with data collected along an arbitrary 3-D radar path. This approach is useful, for example, in military applications where the radar platform may undergo some maneuvers, and in radar astronomy where the relative motion is, at least in part, determined by the natural paths of celestial bodies. The authors show that nonideal platform motion can create phase variations in the data which lead to spatially-varying shifts and blurring. A correction procedure is proposed and demonstrated.

Sidelobe artifact is a common problem in image reconstruction from finite-extent Fourier data. Conventional shift-invariant windows applied to the Fourier data, reduce sidelobe artifacts at the expense of worsened mainlobe resolution. Stankwitz et al. (1995) have suggested spatially-variant apodization (SVA) as a means of reducing the sidelobe artifacts, while preserving the mainlobe resolution. SVA adaptively selects windows from a set of raised-cosine weighting functions. The algorithm is heuristically motivated, and is known to be effective in synthetic aperture radar (SAR) imaging. However, this technique has received only limited analysis. In this paper, we formulate SVA as a spectral estimator, and show that SVA is a special version of the minimum variance spectral estimator (MVSE). We study the properties of SVA that are inherited from MVSE. Then, we consider the application of SVA to spectral estimation and Fourier reconstruction. Although SVA is effective in SAR, we show that it has limitations for reconstructing real-valued extended targets.

We consider the problem of radar imaging of 3-D targets, where conventional 2-D synthetic aperture radar (SAR) techniques are inadequate, because of range-Doppler ambiguities. This problem is cast into the form of a spectral estimation problem, which can be solved using a variety of methods. After giving an overview of two methods, ESPRIT and RELAX, we demonstrate this approach using simulated data.

We consider a simple modification to classical block truncation coding (BTC) that may permit the use of larger block sizes for some applications. although BTC is favored for its simple encoding and decoding, its disadvantage is the relatively low compression ratio achieved. Bit rates on the order of 1 bit/pixel require large block sizes, which generally produce unacceptable degradation. However, these undesirable effects can be reduced considerably by incorporating error diffusion, which is a technique used in halftoning, into the BTC encoding stage.>

Approaching targets can be imaged, with high resolution, using inverse synthetic aperture radar (ISAR). Image formation via the FFT, however, produces an image where the targets are shifted from their true locations. We begin by reviewing the Fourier model of ISAR image formation, which utilizes a Taylor series approximation to the range function. We then show that this Taylor series approximation is equivalent to the straight-line projection assumption in the tomographic model for ISAR. This approximation, plus the presumption that the acquired Fourier data lie on a Cartesian rather than polar grid, are shown to cause the target shifts. We derive formulas for accurately estimating the target shifts. These formulas can provide the basis for effectively compensating the ISAR image for scenarios with small, widely-spaced targets.>

Derives the wavenumber-domain SAR processing algorithm within a Hilbert-space framework. The derivation of the algorithm is motivated from geometrical considerations and the resulting image formation procedure is seen to be optimal. A new derivation of the algorithm, using a direct approximation of an exact Fourier identity, is given. The exact formulation is shown to be equivalent to previous derivations with appropriate approximations. Then, an approximate derivation of the algorithm is given which uses the principle of stationary phase to approximate the Fourier transforms. Fortunately, the approximations are exceedingly good for typical SAR imaging scenarios. Requirements on sampling for discrete implementation of the algorithm are discussed and the resolution achievable using this algorithm is shown to agree with that obtainable in a comparable spotlight scenario.>

The tomographic formulation of spotlight-mode synthetic aperture radar (SAR) is based on a plane-wave assumption that requires the dimension of the target scene to be much smaller than the range to the target. When this assumption is violated, the resulting image exhibits target smearing and target shift. For severely squinted SAR operation, which arises in runway imaging (for aircraft landing applications), the authors have observed that the smearing is not serious, but that targets are shifted considerably from their correct locations, especially for targets near the edges of the scene. They analyze this problem by first deriving an expression for the spatial shift of a reconstructed point reflector for bandpass Fourier data with phase error. Then, they analyze the point target shift in squint-mode SAR, by incorporating the higher-order terms of the range function. Simulation results for isolated point reflectors are presented.>

Considers the application of a spotlight-mode synthetic aperture radar (SAR) imaging technique to the problem of high-resolution lunar imaging and other related radar astronomy problems. This approach offers improved image quality, compared with conventional processing, at the expense of increased computational effort. Results of the processing of lunar data acquired with the 12.6 cm wavelength radar system at Arecibo Observatory are presented, and compared with the best available published result, which uses focusing techniques from strip-mode SAR.>

The authors develop a theory for high-resolution, microwave imaging of approaching targets, using concepts from synthetic aperture radar. Assuming a stepped-frequency waveform, it is shown that the demodulated returned signals provide approximate samples of the two-dimensional Fourier transform of the target reflectivity, on a nearly Cartesian grid. The Fourier model is only an approximation, however, and it is shown that the effect of the true non-Fourier nature of the problem is to shift point targets in the cross-range direction from their actual locations. The importance of autofocus techniques is illustrated, and directions for future research are suggested.>

The problem of reconstructing a bandlimited signal from a set of nonuniformly spaced samples is addressed, with particular emphasis on a method due to J.L. Yen (1956). A simple proof for the derivation of Yen's interpolation formula is given. The interpolating process is then modeled as a linear time-varying system and is analyzed in the frequency domain by means of its bifrequency transmission function (BFTF). It is proven that Yen's interpolator can be obtained as a special case of an optimal frequency-domain design of its BFTF using a weighted least-square criterion. A practical interpolator, based on Yen's method, is considered for situations where the number of samples is large or infinite and the original Yen interpolation formula is inapplicable. The performance of this modified Yen interpolator is compared with that of other commonly used interpolation methods in terms of achievable S/N.>

An algorithm was proposed for computer-aided tomography and spotlight-mode synthetic aperture radar in a recent work by W. Lawton (IEEE Trans. Acoust., Speech, and Signal Process, vol.36, p.933-971, June 1988). This processing scheme assumes that Fourier transform samples of the projections obtained lie on a concentric-squares polar grid. The spatial-domain image is produced through a series of convolutions and DFTs (discrete Fourier transforms), all performed using FFTs (fast Fourier transforms). The algorithm requires O(N/sup 2/logN) operations to produce an N*N image and is made possible by the geometry of the concentric-squares polar grid. The present authors more completely describe the Lawton algorithm and derive a closed-form, approximate expression for its impulse response for the case of spotlight-mode synthetic aperture radar. A formula is given for the exact number of multiplications required by the algorithm, and a set of simulated point target responses is given to help assess the image quality provided by the algorithm.>

Strip-mapping synthetic aperture radar (SAR) is a microwave imaging system capable of producing high resolution terrain images from data collected by a relatively small airborne or spaceborne antenna. Although strip-mapping SAR has been widely covered in the technical literature for at least 25 years, this literature tends to be inaccessible to researchers outside of the radar community due to the following factors: use of unfamiliar terminology, assumption of a prior understanding of basic radar signal processing, and the use of a Doppler viewpoint. In this paper we derive and discuss the fundamental strip-mapping SAR imaging equations from first principles, with radar terminology minimized. The role of the Doppler effect is clarified. Although this paper is fairly elementary, it is hoped that it might serve as a bridge for the signal processing community to the existing literature on strip-mapping SAR.

In this paper we consider the effect of median prefiltering on the subsequent estimation and detection of edges in digital images. Where possible, a quantitative statistical comparison is made for a number of filters defined with two-dimensional geometries; in some cases one-dimensional analyses are required to illustrate certain points. Noise images prefiltered by median filters defined with a variety of windowing geometries are used to support the analysis, and it is found that median prefiltering improves the performance of both thresholding and zero-crossing based edge detectors.

We investigate the usefulness of support-limited extrapolation as a tool for improving the estimate of the magnitude of a finite extent complex-valued signal from samples of its Fourier transform that are offset from the origin. The motivating application is spotlight mode synthetic aperture radar where the complex-valued signal to be recovered is microwave reflectivity and the measured Fourier samples of this reflectivity constitute the signal to be extrapolated. Both analytical and simulation results are presented supporting an earlier conjecture that extrapolation will be of little benefit unless the width of the known Fourier data is less than or equal to the bandwidth of just the magnitude of the reflectivity.

Optimal statistical procedures are formulated for detecting object boundaries in speckle noise imagery. Although speckle noise is found in many applications, our principal interest is in synthetic aperture radar (SAR). The first procedure described is parametric. The ratio of local neighborhood averages is thresholded, which is a statistically natural approach since the noise is often modeled as multiplicative. A nonparametric procedure based on a linear rank statistic is also described, which can be shown to be locally most powerful (among rank tests). Examples are given, and comparisons are offered. The parametric scheme performed slightly better than the rank-order method in the examples, but the inherent robustness of the latter may recommend it for the practical application.

Several well known imaging techniques operate by recording samples of the Fourier transform of the object function and then reconstructing the object function by means of the 2D inverse FFT. A central problem arises in interpolating from the inherent polar raster to a rectangular raster, so the inverse FFT can be properly applied. This paper considers the use of nearest neighbor and inverse distance interpolation when the angular recording interval is relatively small. The objective is to obtain a computationally simple reconstruction algorithm that achieves good resolution in the final image plane.

We compare the performance of three algorithms for polar-to-cartesian interpolation in spotlight mode synthetic aperture radar. The original polar data in this case lies far from the origin in a region of very small angular extent.

Motivated by the ability of synthetic aperture radar and related imaging systems to produce images of suprisingly high quality, we consider the problem of reconstructing the magnitude of a complex signalffrom samples of the Fourier transform offlocated in a small region offset from the origin. It is shown that high quality speckle reconstructions are possible so long as the phase offis highly random. In this case the quality of the reconstruction is insensitive to the location of the known Fourier data, and edges at all orientations are reproduced equally well. Computer examples are presented demonstrating these attributes.

In this paper we consider two new techniques for image restoration, using order-constrained least-squares methods. The first technique consists of a cross-shaped moving window, within which two operations are combined. The first operation consists of simple hypothesis tests for monotonicity in both the horizontal and vertical directions. The second step finds the best least-squares fit of the input variates in both directions, constrained by the results of the hypothesis tests. The second technique consists of a square moving window, again combining two operations. With the first operation, we introduce a new edge detector with specific edge height δ. Based on detection or non-detection of an edge, we either apply order-constrained least-squares methods to determine the output, or simply average. The techniques described are applied to an actual noise-corrupted image.

In this paper we consider a nonlinear filter whose output is given by a linear combination of the order statistics of the input sequence. Assuming a constant signal in white background noise, the coefficients in the linear combination are chosen to minimize the output MSE for several noise distributions. This new general filter is superior to the well-known median filter, since the median is just a special case.

The required sampling rate is an important consideration in digital implementations of analog shift-variant systems. Sufficient sampling rates have been previously derived. In this paper we derive the necessary rate and illustrate the method for its computation.

It is shown that the problem of determining the maximum amplitude limit cycle of a given period in a second order digital filter can be formulated as an integer linear program. Unfortunately, for even moderate limit cycle periods, the ILP can require considerable CPU time. A more straightforward procedure is thus proposed, which considers all possible initial states, which can be a maximum amplitude state of a limit cycle. Results are presented which significantly reduce the number of states requiring consideration, thereby resulting in a computationally feasible algorithm.

The prime factor FFT

Errors due to finite wordlength can in general be quite large in narrowband recursive digital filters. It has been shown that the mean squared roundoff error may be considerably reduced by using a technique known as error spectrum shaping (ESS), which utilizes quantizer feedback. In this paper it is shown that ESS can also significantly reduce the amplitude of zero input limit cycle oscillations. A more complete analysis of the MSE is then presented, which is useful for design purposes.